Who Stole the Tarts? Part VII

"Well," said the King, "here is your sugar, so you can make me the tarts."

"Without salt?" asked the Queen.

So! The salt had also been stolen! Well, this time it was found that the culprit was either the Caterpillar, Bill the Lizard, or the Cheshire Cat. (One of them had come into the kitchen and eaten up all the salt; the container wasn't missing.) The three were tried and made the following statements in court:

CATERPILLAR: Bill the Lizard ate the salt.
BILL THE LIZARD: That is true!
CHESHIRE CAT: I never ate the salt!

As it happened, at least one of them lied and at least one told the truth.
Who stole the salt?

(Source: Alice in Puzzle Land: A Carrollian Tale for Children Under Eighty by Raymond Smullyan)

Initially, it seems like there could be a total of \(\dbinom{3}{1}\) ways that one truth-teller could be assigned to a group of two liars and, then, respectively, \( \dbinom{3}{2}\) ways two truth-tellers could be assigned to a liar, to create the all the groups with compositions acceptable by the last stipulation in the story. Both of these combinatorial expressions evaluate to three and so six possibilities all told would appear to need consideration.

But wait! Before I enumerate them all! Know that things are actually more closely constrained because of the agreement between the Caterpillar and Bill the Lizard. There are indeed only two possibilities, namely:

• Caterpillar is truthful; Bill the Lizard is truthful; Cheshire Cat is lying
• Caterpillar is lying; Bill the Lizard is lying; Cheshire Cat is truthful

In the first instance, Bill the Lizard ate the salt. But then it is the case that the Cheshire Cat is not lying. Only the second option is therefore tenable and, in this instance, the Caterpillar and Bill the Lizard are both lying and the Cheshire Cat is truthful. From this it can be inferred that neither the Cheshire Cat nor Bill the Lizard ate up all of the salt, and the only remaining option is that the Caterpillar did it.